Finite Group Databasebeta

← #13327698

Subgroups — #13327698

Order 1792.

1282 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#537C211yesmaximal, characteristic, complemented
2#11740070C212yesmaximal, complemented
3#11739644C211yesmaximal, characteristic, complemented
4#11729632C214yesmaximal, complemented
5#11729628C214yesmaximal, complemented
6#11729429C212yesmaximal, complemented
7C2×#89323C211yesmaximal, characteristic, complemented
8C22×C7×#41—21no—
9C7×#537—24no—
10C7×#537—22no—
11C7×#537C2214yes—
12C2×C7×C22⋊Q8—22no—
13C2×C7×#121—22no—
14C2×C7×#121—21no—
15C2×C7×#121C2212yes—
16C22×C7×C4⋊C4C2211yescharacteristic
17C2×#6493C2211yescharacteristic
18C2×#6492—21no—
19#89989—22no—
20#89985—21no—
21#89984C2211yescharacteristic
22C2×C22⋊(C7⋊C8)C2212yes—
23#89810—24no—
24#89808C2214yescomplemented
25#89805C2214yescomplemented
26#89766—22no—
27#89763C2212yescomplemented
28#89755C2212yescomplemented
29#89739—22no—
30#89332C2218yescomplemented
31#89323C2214yescomplemented
32#18292—71nomaximal, Sylow
33C2×C7×#41—24no—
34C2×C7×#41—24no—
35C2×C7×#41—44no—
36C2×C7×#41—44no—
37C22×C7×Q8—21no—
38C22×C7×D4—21no—
39C22×C7×D4—22no—
40C22×C7×D4—21no—
41C22×C7×D4D412yes—
42C23×C4×C7—22no—
43C23×C4×C7D411yescharacteristic
44C23×C4×C7D411yescharacteristic
45C23×C4×C7C2311yescharacteristic
46C22×C22⋊D7—21no—
47C22×D28C2311yescharacteristic
48C7×C22⋊Q8—28no—
49C7×C22⋊Q8—28no—
50C7×#121—24no—
51C7×#121—28no—
52C7×#121—28no—
53C7×#121—24no—
54C7×#121—22no—
55C7×#121—22no—
56C7×#121D414yes—
57C7×#121C2314yes—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4—22no—
60C2×C7×C4⋊C4—22no—
61C2×C7×C4⋊C4D412yes—
62C2×C7×C4⋊C4D412yes—
63C2×C7×C4⋊C4C2312yes—
64C2×C7×C22⋊C4—22no—
65C2×C7×C22⋊C4—22no—
66C2×C22⋊Dic7—21no—
67C2×#1141—24no—
68#6493—22no—
69#6493—22no—
70#6492—44no—
71C2×(C2×D7)⋊C4—22no—
72C2×(C2×D7)⋊C4—21no—
73C2×(C2×D7)⋊C4—22no—
74C2×C4⋊Dic7—21no—
75C2×C4⋊Dic7C2311yescharacteristic
76C2×Dic7⋊C4—22no—
77C22×C7⋊C8—22no—
78#6406—24no—
79C7⋊#95—22no—
80C7⋊#95C2312yes—
81C22⋊(C7⋊C8)—22no—
82C22⋊(C7⋊C8)C2314yes—
83#6383—24no—
84#6381—24no—
85C2×#537D711yescharacteristic, complemented
86#3186—72no—
87#2618—71no—
88#1610—74no—
89#1598—74no—
90#1519—72no—
91C2×#324—71no—
92C24×C7—22no—
93C24×C7C2×D411yescharacteristic
94C23×D7—21no—
95C7×#41—48no—
96C7×#41—416no—
97C7×#41—48no—
98C2×C7×Q8—24no—
99C2×C7×Q8—44no—
100C2×C7×D4—24no—
101C2×C7×D4—24no—
102C2×C7×D4—24no—
103C2×C7×D4—24no—
104C2×C7×D4—22no—
105C2×C7×D4—44no—
106C2×C7×D4—44no—
107C2×C7×D4—44no—
108C2×C7×D4—82no—
109C2×C7×D4—82no—
110C2×C7×D4C2×D414yes—
111C22×C4×C7—21no—
112C22×C4×C7—21no—
113C22×C4×C7—22no—
114C22×C4×C7—28no—
115C22×C4×C7—21no—
116C22×C4×C7—22no—
117C22×C4×C7—21no—
118C22×C4×C7—48no—
119C22×C4×C7—81no—
120C22×C4×C7—81no—
121C22×C4×C7D814yescomplemented
122C22×C4×C7C2×D412yes—
123C22×C4×C7C2×D412yes—
124C22×C4×C7C2×D411yescharacteristic
125C22×C4×C7C2×D412yes—
126C22×C4×C7C2×D412yes—
127C22×C4×C7C2411yescharacteristic
128C2×C22⋊D7—41no—
129C2×C22⋊D7—41no—
130C2×C22⋊D7—42no—
131C2×C22⋊D7—42no—
132C22×Dic7—21no—
133C22×Dic7—22no—
134C2×D28—22no—
135C2×D28—81no—
136C7×C4⋊C4—24no—
137C7×C4⋊C4—24no—
138C7×C4⋊C4—22no—
139C7×C4⋊C4—24no—
140C7×C4⋊C4C2×D414yes—
141C7×C22⋊C4—28no—
142C7×C22⋊C4—24no—
143C7×C22⋊C4—24no—
144C22⋊Dic7—42no—
145#1141—84no—
146(C2×D7)⋊C4—42no—
147(C2×D7)⋊C4—44no—
148(C2×D7)⋊C4—82no—
149C4⋊Dic7—22no—
150C4⋊Dic7—81no—
151Dic7⋊C4—44no—
152C2×C7⋊C8—24no—
153C2×C7⋊C8—82no—
154C22×#41—21no—
155#537—24no—
156#537—22no—
157#537C2×D714yes—
158C2×#123—141no—
159C2×C22⋊Q8—22no—
160C2×#121—22no—
161C2×#121—21no—
162C2×#121—71no—
163C2×#121C2×D712yes—
164C22×C4⋊C4C2×D711yescharacteristic
165#480—74no—
166#480—142no—
167#466—144no—
168#447—74no—
169#447—142no—
170C2×C22⋊C8—72no—
171#398—72no—
172#396—142no—
173#394—72no—
174#386—141no—
175#380—71no—
176#327—78no—
177#324—74no—
178C23×C7—24no—
179C23×C7—41no—
180C23×C7—41no—
181C23×C7—44no—
182C23×C7—44no—
183C23×C7—44no—
184C23×C7#12011yescharacteristic
185C23×C7#12011yescharacteristic
186C23×C7C2×D812yes—
187C23×C7AGL(1,ℤ/8)12yes—
188C23×C7C22×D411yescharacteristic
189C22×D7—42no—
190C22×D7—42no—
191C22×D7—41no—
192C22×D7—41no—
193C22×D7—41no—
194C7×Q8—48no—
195C7×D4—44no—
196C7×D4—48no—
197C7×D4—44no—
198C7×D4—44no—
199C7×D4—48no—
200C7×D4—82no—
201C7×D4—84no—
202C2×C4×C7—22no—
203C2×C4×C7—22no—
204C2×C4×C7—24no—
205C2×C4×C7—24no—
206C2×C4×C7—22no—
207C2×C4×C7—24no—
208C2×C4×C7—42no—
209C2×C4×C7—41no—
210C2×C4×C7—41no—
211C2×C4×C7—42no—
212C2×C4×C7—416no—
213C2×C4×C7—42no—
214C2×C4×C7—44no—
215C2×C4×C7—42no—
216C2×C4×C7—81no—
217C2×C4×C7#12012yes—
218C2×C4×C7C2×D814yes—
219C2×C4×C7C22×D412yes—
220C22⋊D7—82no—
221C22⋊D7—82no—
222C2×Dic7—42no—
223C2×Dic7—42no—
224C2×Dic7—41no—
225C2×Dic7—41no—
226C2×Dic7—42no—
227C2×Dic7—41no—
228D28—44no—
229C7⋊C8—44no—
230C2×#41—24no—
231C2×#41—24no—
232C2×#41—44no—
233C2×#41—44no—
234C22×Q8—21no—
235C22×D4—21no—
236C22×D4—22no—
237C22×D4—21no—
238C22×D4—71no—
239C22×D4—141no—
240C22×D4C22⋊D712yes—
241C23×C4—22no—
242C23×C4C22⋊D711yescharacteristic
243C23×C4C22⋊D711yescharacteristic
244C23×C4C22×D711yescharacteristic
245C2×D8—144no—
246C22×C8—142no—
247#123—284no—
248C22⋊Q8—28no—
249C22⋊Q8—28no—
250#121—24no—
251#121—28no—
252#121—28no—
253#121—24no—
254#121—22no—
255#121—22no—
256#121—142no—
257#121—142no—
258#121C22⋊D714yes—
259#121C22×D714yes—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—71no—
264C2×C4⋊C4—142no—
265C2×C4⋊C4—141no—
266C2×C4⋊C4C22⋊D712yes—
267C2×C4⋊C4C22⋊D712yes—
268C2×C4⋊C4C22×D712yes—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—141no—
272C2×C22⋊C4—141no—
273C2×C22⋊C4—142no—
274C2×C22⋊C4—142no—
275#107—144no—
276#102—144no—
277#102—144no—
278C22⋊C8—74no—
279C22⋊C8—142no—
280#95—72no—
281#95—142no—
282C22×C7—21no—
283C22×C7—21no—
284C22×C7—21no—
285C22×C7—21no—
286C22×C7—21no—
287C22×C7—21no—
288C22×C7—42no—
289C22×C7—42no—
290C22×C7—44no—
291C22×C7—44no—
292C22×C7—44no—
293C22×C7—88no—
294C22×C7#44712yes—
295C22×C7#44712yes—
296C22×C7C2×#12011yescharacteristic
297C22×C7C22×D811yescharacteristic
298C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
299C2×D7—82no—
300C2×D7—82no—
301C2×D7—81no—
302C2×D7—81no—
303C2×D7—81no—
304C4×C7—24no—
305C4×C7—22no—
306C4×C7—22no—
307C4×C7—44no—
308C4×C7—42no—
309Dic7—82no—
310Dic7—81no—
311C24—22no—
312C24—141no—
313C24C2×C22⋊D711yescharacteristic
314#41—48no—
315#41—416no—
316#41—48no—
317C2×Q8—24no—
318C2×Q8—44no—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—22no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—82no—
328C2×D4—82no—
329C2×D4—142no—
330C2×D4—282no—
331C2×D4—282no—
332C2×D4—281no—
333C2×D4—281no—
334C2×D4—561no—
335C2×D4C2×C22⋊D714yes—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—28no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—48no—
344C22×C4—81no—
345C22×C4—81no—
346C22×C4—142no—
347C22×C4—141no—
348C22×C4#114114yescomplemented
349C22×C4C2×C22⋊D712yes—
350C22×C4C2×C22⋊D712yes—
351C22×C4C2×C22⋊D711yescharacteristic
352C22×C4C2×C22⋊D712yes—
353C22×C4C2×C22⋊D712yes—
354C22×C4C23×D711yescharacteristic
355D8—564no—
356C2×C8—144no—
357C2×C8—562no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—24no—
362C4⋊C4—142no—
363C4⋊C4—284no—
364C4⋊C4—561no—
365C4⋊C4C2×C22⋊D714yes—
366C22⋊C4—28no—
367C22⋊C4—24no—
368C22⋊C4—24no—
369C22⋊C4—282no—
370C22⋊C4—284no—
371C22⋊C4—282no—
372C22⋊C4—562no—
373C2×C7—21no—
374C2×C7—21no—
375C2×C7—22no—
376C2×C7—22no—
377C2×C7—84no—
378C2×C7C2×#44711yescharacteristic
379C2×C7#302011yescharacteristic
380C2×C7#302611yescharacteristic
381D7—161no—
382C23—24no—
383C23—41no—
384C23—41no—
385C23—44no—
386C23—44no—
387C23—44no—
388C23—282no—
389C23—282no—
390C23—281no—
391C23—281no—
392C23—281no—
393C23C2×#114112yes—
394C23#649512yes—
395C23#651611yescharacteristic
396C23#651611yescharacteristic
397C23C22×C22⋊D711yescharacteristic
398Q8—48no—
399D4—44no—
400D4—48no—
401D4—44no—
402D4—44no—
403D4—48no—
404D4—82no—
405D4—84no—
406D4—284no—
407D4—562no—
408D4—562no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—416no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—81no—
424C2×C4—282no—
425C2×C4—282no—
426C2×C4—282no—
427C2×C4—281no—
428C2×C4—281no—
429C2×C4—281no—
430C2×C4C2×#114114yes—
431C2×C4#651612yes—
432C2×C4C22×C22⋊D712yes—
433C8—284no—
434C7#1829211yescharacteristic, complemented, Sylow
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—42no—
442C22—42no—
443C22—44no—
444C22—44no—
445C22—44no—
446C22—88no—
447C22—562no—
448C22—562no—
449C22—561no—
450C22—561no—
451C22—561no—
452C22#8998912yes—
453C22#8998912yes—
454C22C22×#114111yescharacteristic
455C22C2×#649511yescharacteristic
456C22C2×#651611yescharacteristic
457C4—24no—
458C4—22no—
459C4—22no—
460C4—44no—
461C4—42no—
462C4—562no—
463C4—561no—
464C2—21no—
465C2—22no—
466C2—21no—
467C2—22no—
468C2—84no—
469C2—1121no—
470C2C2×#8998911yescharacteristic
471C2#1174155511yescharacteristic
472C2#1174156111yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#537C2
socleC22×C7C2×#120
hypercenterC2×#537D7
hypocenterC7#18292
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#537

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×C7×#537C2semidirect
2. #11740070C2semidirect
3. #11739644C2semidirect
4. #11729632C2semidirect
5. #11729628C2semidirect
6. #11729429C2semidirect
7. C2×#89323C2semidirect
24. #89808C22semidirect
25. #89805C22semidirect
27. #89763C22semidirect
28. #89755C22semidirect
30. #89332C22semidirect
31. #89323C22semidirect
85. C2×#537D7semidirect
121. C22×C4×C7D8semidirect
348. C22×C4#1141semidirect
434. C7#18292semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C7×#5372
#117400702
#117396442
#117296322
#117296282
#117294292
C2×#893232
#182927